basically all you need to do is apply the remainder theorem. So all you do is sub in x=1 into the equation so you get:
(1)^4 + p(1)^3 +q(1) + 1
= 1+p+q+1
= p+q+2
and we know that the remainder is 3 i.e. Q(x) = 3 therefore p+q+2 = 3...(1)
Do the same thing but now sub in x= -2 to get:
-8p-2q+17. we know that the remainder is -3 i.e. Q(x)= -3 therefore -8p-2q+17=-3...(2)
Then you just solve these two equations simultaneously